Can someone solve this, i have tried for 2 days!!

[obelisk1151]

Unfortunately its on german and my english ins't that good so i've translated it over google translate:
Given are the 2 × 3 matrices A1 = (1 0 1 0 1 2), A2 = (1 0 1 3 1 0),A3 = (5 0 5 6 5 6).
(a) Are these matrices of R2 × 3 linearly dependent? If yes, represent A3 as a linear combination of the other two.
(b) Does the span of the matrices {A1, A2, A3} form a subspace of R2 × 3? If so, what is the dimension of this subspace?
(c) What is the dimension of the R-vector space R2 × 3? Enter a base.
(d) Which dimension has the R-vector space C2 × 3? Enter a base.


I have already done the (a) part, but i don't know how do to the rest, pls someone help me.

 

[Deleted member 4993]

View attachment 14732
Unfortunately its on german and my english ins't that good so i've translated it over google translate:
Given are the 2 × 3 matrices A1 = (1 0 1 0 1 2), A2 = (1 0 1 3 1 0),A3 = (5 0 5 6 5 6).
(a) Are these matrices of R2 × 3 linearly dependent? If yes, represent A3 as a linear combination of the other two.
(b) Does the span of the matrices {A1, A2, A3} form a subspace of R2 × 3? If so, what is the dimension of this subspace?
(c) What is the dimension of the R-vector space R2 × 3? Enter a base.
(d) Which dimension has the R-vector space C2 × 3? Enter a base.


I have already done the (a) part, but i don't know how do to the rest, pls someone help me.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING

Please share your work/thoughts about this assignment.

Please show us the work you have done to solve part (a).

 

[Steven G]

Well you got me curious. Are the three matrices independent or not? Don't you think that answer would help for some/all the other parts.

 

[obelisk1151]

Well you got me curious. Are the three matrices independent or not? Don't you think that answer would help for some/all the other parts.

I wanted to sent the work ive done (a), but after checking it i have realized that i've screwed up.

 

[obelisk1151]

Well you got me curious. Are the three matrices independent or not? Don't you think that answer would help for some/all the other parts.

Now i dont even know how to do (a)

 

[HallsofIvy]

The first question asks whether $\displaystyle \begin{pmatrix}5 & 0 & 5 \\ 6 & 5 & 6\end{pmatrix}$ is a linear combination of $\displaystyle \begin{pmatrix}1 & 0 & 1\\ 0 & 1 & 2\end{pmatrix}$ and $\displaystyle \begin{pmatrix}1 & 0 & 1\\ 3 & 1 & 0\end{pmatrix}$.

That is, do there exist a and b such that $\displaystyle a\begin{pmatrix}1 & 0 & 1\\ 0 & 1 & 2\end{pmatrix}+ b\begin{pmatrix}1 & 0 & 1\\ 3 & 1 & 0\end{pmatrix}= \begin{pmatrix}a+b & 0 & a+b \\ 3a & a+b & 2b\end{pmatrix}= \begin{pmatrix}5 & 0 & 5 \\ 6 & 5 & 6\end{pmatrix}$.

And that is the same as asking whether there exist a and b such that a+ b= 5, 3a= 6, 2b= 6. If there are what are a and b?

But it is easy to see that the first two matrices are linearly independent since one is not a multiple of the other. So the subspace spanned by those two matrices is 2 dimensional, all such matrices being of the form $\displaystyle a\begin{pmatrix}1 & 0 & 1\\ 0 & 1 & 2\end{pmatrix}+ b\begin{pmatrix}1 & 0 & 1\\ 3 & 1 & 0\end{pmatrix}$.

"$\displaystyle R_{2x3}$", the space of all 2 by 3 matrices, consists of all matrices of the form $\displaystyle \begin{pmatrix} a & b & c \\ d & e & f\end{pmatrix}$. That contains 6 independent numbers so is of dimension 6. Find each of the 6 basis matrices by taking each of a, b, c, d, e, and f to be 1 and the others 0. Two such basis matrices are $\displaystyle \begin{pmatrix}1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$ and $\displaystyle \begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}$

For the last one, every complex number can be constructed from two real numbers so $\displaystyle C_{23}$ has dimension 2(6)= 12. There are 12 basis matrices. Two of them are Two such basis matrices are $\displaystyle \begin{pmatrix}1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$ and $\displaystyle \begin{pmatrix}i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$.

 

[Steven G]

I wanted to sent the work ive done (a), but after checking it i have realized that i've screwed up.

Send the work so we can see where you went wrong so we can help you.

 

[obelisk1151]

The first question asks whether $\displaystyle \begin{pmatrix}5 & 0 & 5 \\ 6 & 5 & 6\end{pmatrix}$ is a linear combination of $\displaystyle \begin{pmatrix}1 & 0 & 1\\ 0 & 1 & 2\end{pmatrix}$ and $\displaystyle \begin{pmatrix}1 & 0 & 1\\ 3 & 1 & 0\end{pmatrix}$.

That is, do there exist a and b such that $\displaystyle a\begin{pmatrix}1 & 0 & 1\\ 0 & 1 & 2\end{pmatrix}+ b\begin{pmatrix}1 & 0 & 1\\ 3 & 1 & 0\end{pmatrix}= \begin{pmatrix}a+b & 0 & a+b \\ 3a & a+b & 2b\end{pmatrix}= \begin{pmatrix}5 & 0 & 5 \\ 6 & 5 & 6\end{pmatrix}$.

And that is the same as asking whether there exist a and b such that a+ b= 5, 3a= 6, 2b= 6. If there are what are a and b?

But it is easy to see that the first two matrices are linearly independent since one is not a multiple of the other. So the subspace spanned by those two matrices is 2 dimensional, all such matrices being of the form $\displaystyle a\begin{pmatrix}1 & 0 & 1\\ 0 & 1 & 2\end{pmatrix}+ b\begin{pmatrix}1 & 0 & 1\\ 3 & 1 & 0\end{pmatrix}$.

"$\displaystyle R_{2x3}$", the space of all 2 by 3 matrices, consists of all matrices of the form $\displaystyle \begin{pmatrix} a & b & c \\ d & e & f\end{pmatrix}$. That contains 6 independent numbers so is of dimension 6. Find each of the 6 basis matrices by taking each of a, b, c, d, e, and f to be 1 and the others 0. Two such basis matrices are $\displaystyle \begin{pmatrix}1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$ and $\displaystyle \begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}$

For the last one, every complex number can be constructed from two real numbers so $\displaystyle C_{23}$ has dimension 2(6)= 12. There are 12 basis matrices. Two of them are Two such basis matrices are $\displaystyle \begin{pmatrix}1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$ and $\displaystyle \begin{pmatrix}i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$.

a) didn't the first question ask me if all 3 matrices are linearly dependent?
and if they are then to represent a3 as a linear combination of a2 and a1
b) its asking if the span of all 3 matrices form a subspace of R2+3 and what dimension that subspace has.

 

[Steven G]

a) didn't the first question ask me if all 3 matrices are linearly dependent?
and if they are then to represent a3 as a linear combination of a2 and a1

Yes, you are correct and HallsofIvy did just that. Halls showed that aA₁ + bA₂ = A₃ under certain constraints on a and b. Were you able to solve for a and b?

 

[obelisk1151]

Yes, you are correct and HallsofIvy did just that. Halls showed that aA₁ + bA₂ = A₃ under certain constraints on a and b. Were you able to solve for a and b?

wait im confused i know that to prof that that they are linearly dependent its xA1+yA2+zA3=0 and if x,y,z are all 0 then the matrices are linearly independent otherwise if x,y,z are not all 0 then the matrices are linearly dependent
or im i wrong here?
aA1 + bA2 = A3 is to represent A3 as a linear combination of a2 and a1

 

[Steven G]

wait im confused i know that to prof that that they are linearly dependent its xA1+yA2+zA3=0 and if x,y,z are all 0 then the matrices are linearly independent otherwise if x,y,z are not all 0 then the matrices are linearly dependent
or im i wrong here?
aA1 + bA2 = A3 is to represent A3 as a linear combination of a2 and a1

You are correct that to show that they are linearly dependent you can conclude that xA1+yA2+zA3=0 and x, y and z are NOT all 0.
But is that the only way? If xA1+yA2+zA3=0 doesn't that imply (if z$\displaystyle \neq$0) A3 = (-x/z)A1 + (-y/z)A2????

 

[obelisk1151]

You are correct that to show that they are linearly dependent you can conclude that xA1+yA2+zA3=0 and x, y and z are NOT all 0.
But is that the only way? If xA1+yA2+zA3=0 doesn't that imply (if z$\displaystyle \neq$0) A3 = (-x/z)A1 + (-y/z)A2????

alright but what about b)? can you explain that?

 

[Steven G]

alright but what about b)? can you explain that?

Does the span of the matrices {A1, A2, A3} form a subspace of R2 × 3? If so, what is the dimension of this subspace?
First you need to know what it means for a set of matrices to span all 2X3 matrices. It means any 2x3 matrix, A, can be expressed as a linear combination of A1, A2 and A3. Try this and show us your work.

Oh, and did you do the work for part a? Please show that first?

 

[obelisk1151]

Does the span of the matrices {A1, A2, A3} form a subspace of R2 × 3? If so, what is the dimension of this subspace?
First you need to know what it means for a set of matrices to span all 2X3 matrices. It means any 2x3 matrix, A, can be expressed as a linear combination of A1, A2 and A3. Try this and show us your work.

Oh, and did you do the work for part a? Please show that first?

i did part a) and i got that -5z+5z=0,and it confused me
i did it with xA1+yA2+zA3=0

 

[Steven G]

i did part a) and i got that -5z+5z=0,and it confused me
i did it with xA1+yA2+zA3=0

Please show us your work. How many times do you have to be asked?

 

[obelisk1151]

Please show us your work. How many times do you have to be asked?

 

[Steven G]

View attachment 14744

Well why did you stop there?
-5z + 5z =0, So you have 0 = 0. What does that tell you? It should tell you that Z can be any number. Now finish up by figuring out what x and y equal and they my be in terms of z!
Continue!

 

[obelisk1151]

Well why did you stop there?
-5z + 5z =0, So you have 0 = 0. What does that tell you? It should tell you that Z can be any number. Now finish up by figuring out what x and y equal and they my be in terms of z!
Continue!

i dont know how to do it if z can be any number then how can i figure out y an x

 

[Steven G]

i dont know how to do it if z can be any number then how can i figure out y an x

Let z = k for some real number k
Then in terms of k what does x and y equal?

For example if r=3t, s=-2t and t=k, then the solution will be, r=3k, s=-2k and t=k. This means that there are an infinite number of solutions. So give us your final answer. That is tell us if the 3 matrices are independent or not and if they are dependent give us the linear combination that shows that.

 

[obelisk1151]

but because there are an infinite number of solutions that means then that the matrices are dependent right?